X-ray Powder Diffraction II ray Powder Diffraction II
Peak Intensities Peak Intensities
Chemistry 754
Solid State Chemistry
Lecture #10
Outline Outline
• Elastic scattering of X-rays by atoms
• Diffraction peak intensities
– Structure Factors
– Multiplicity
– Lorentz and Polarization Factors
– Temperature Factors
• Neutron diffraction
• Electron diffraction
2
Interaction of X Interaction of X-rays with Matter rays with Matter
These are the
diffracted X-rays
Elastic Scattering by an Electron Elastic Scattering by an Electron
• Charged particles (electron) scatter
electromagnetic radiation (x-rays)
– The varying electric field of the X-ray induces an oscillation
of the electron
– The oscillating electron then acts as a source of
electromagnetic radiation
– In this way the x-rays are scattered in all directions
• JJ Thompson analyzed the scattering and found
that:
I = I0[(?0/4p)2(e4/m2r2)sin2?]
I = I0(K/r2) sin2?
? is the angle between the scattering direction and the
direction in which the electron is accelerated
r is the distance from the scattering electron
K is a constant
3
Scattering by an Atom Scattering by an Atom
We can consider an atom to be a collection of
electrons. The electrons around an atom scatter
radiation in the manner described by Thompson.
However, due to the coherence of the radiation we
need to consider interference effects from different
electrons within an atom. This leads to a strong
anglular dependence of the scattering. We express
the scattering power of an atom by its form factor (f).
X-ray Form Factors ray Form Factors
0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6
Sin(?)/?
Form Factor, f
Ca
Ca2+
The form factor is
equivalent to the atomic
number at ? = 0
The form factor
drops rapidly as a
function of (sin ?)/?
4
The Effect of Form Factors on The Effect of Form Factors on
Diffraction Patterns Diffraction Patterns
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5 25 45 65
2-Theta (Degrees)
Intensity
TbBaFe2O5 - 300 K
Synchrotron X-ray
10 30 50 70 90 110 130
2-Theta (Degrees)
Intensity (Arb. Units)
TbBaFe2O5 - 70 K
Neutron Data
The peak intensities drop off at
high angles in an X-ray diffraction
pattern because the form factor
decreases
Neutrons are scattered from the
nucleus and the form factor is not
angle dependent. Intensities do
not drop off at high angle.
Diffraction Intensities Diffraction Intensities
The integrated intensity (peak area) of each powder
diffraction peak is given by the following expression:
I( hkl) = |S( hkl)|2 × Mhkl × LP(?) × TF(?)
– S( hkl) = Structure Factor
– Mhkl = Multiplicity
– LP(?) = Lorentz & Polarization Factors
– TF(?) = Temperature factor (more correctly
referred to as the displacement parameter)
This does not include effects that can sometimes by
problematic such as absorption, preferred orientation
and extinction.
5
Structure Factor Structure Factor
The structure factor reflects the interference between atoms in
the basis (within the unit cell). All of the information regarding
where the atoms are located in the unit cell is contained in the
structure factor. The structure factor is given by the following
summation over all atoms (from 1 to j) in the unit cell:
S( hkl) = ?j fj exp {-i2?(hxj+kyj+lzj)}
– fj = form factor for the jth atom
– h, k & l = Miller indices of the hkl reflection
– xj, yj & zj = The fractional coordinates of the jth atom
To evaluate the value of the imaginary term in the exponential
function, remember Euler’s equation:
exp (-ix) = cos(x) – i sin(x)
The value of this function is a real number when x is a multiple of
2?. It is equal to 1 for even multiples of 2? and -1 for odd
multiples of 2?.
Structure Factors: Example Structure Factors: Example CsCl CsCl
• Let’s calculate the structure factors for the first 6
peaks CsCl. To do this we need to know the atomic
positions and the Miller Indices.
Cl at (0,0,0)
Cs at (½,½,½)